\chapter{Landscape Evolution under river dynamics}
\label{chap:landscapeEvolution}
\begin{flushright}{\slshape    
Whenever a theory appears to you as the only possible one,\\
take this as a sign that you have neither understood the theory\\
nor the problem which it was intended to solve} \\ \medskip
--- \citeauthor{popper:1979}, \citetitle{popper:1979},
\citeyear{popper:1979}
\end{flushright}

This chapter is intended to make the reader familiar with the
topic of landscape evolution under river dynamics and the
processes it involves, in terms of matter transportation and
energy transformation. There is the description and the comparison
of the main approaches adopted in scientific and technical
disciplines for understanding and reproducing this phenomenon.
Before concluding the chapter, the methodology adopted for the
work described within this thesis is briefly introduced.
%%%%%%%%%%%%%%%%%%%%%%%
\section{Landscape and rivers evolution}
\label{sec:basicTheory}
As \graffito{Basic theoretical contents are given to depict the
frame around this work.} it was already explained in the
introduction, the objective of this thesis is to approach the
complex phenomena of landscape and river evolution through a
simulation model based on optimality criteria.
However, before focusing the attention on the model it is necessary
to provide some basic knowledge about the phenomena themselves, in
order to better understand the basic hypothesis and the features
of the model.

A clear definition of landscape evolution is given by traditional
geomorphology discipline, which states:
\blockcquote[see][p. 247]{pazzaglia:2003}{landscape evolution
describes exclusively time-dependent changes from rugged youthful
topography, through the rounded hillslopes of maturity, to death
as a flat plain.}
It appears, from this definition, that the process of landscape
evolution covers multiple scales, both in terms of space and time,
during which the landscape modifies its shape, and therefore mass
distribution, going through different chronological phases. It is
a dynamic process that develops over geologic and human time
scales too and it is driven by multiple agents. In fact, as
\citeauthor{pazzaglia:2003} highlights in his
\citeyear{pazzaglia:2003} paper, quoting a previous paper by
Davis, many processes including \blockcquote[see][p.
251]{pazzaglia:2003}{surface wash, ground water, temperature
change, freeze-thaw, chemical weathering, root and animal
bioturbation} contribute to shape the Earth landscape. However, it
is possible to say that the following three factors are the most
important categories of drivers:
\begin{itemize}
  \item tectonic activities;
  \item interaction between soil and climatic and atmospheric agents;
  \item interaction between terrain and water fluxes.
\end{itemize}

Since the aim of this thesis is studying the just mentioned
process of landscape evolution under the effects of river
dynamics, the analysis is not going to deepen in detail the other
factors shaping the landscape and our attention will be focused on
the interaction between river networks and the terrain.
It becomes then important to characterize river networks and their
basins in relationship with the terrain they develop on and, for
this purpose, \citeauthor{rodriguez:1992}'s words look appropriate: 
\blockcquote[see][p. 1095]{rodriguez:1992}{well-developed river
basins are made up of two interrelated systems: the channel
networks and the hillslopes [\ldots] Hillslopes are the
runoff-producing elements which the network connects}.
Considering that, it emerges that river networks formation and
landscape evolution processes are strongly interdependent,
therefore they cannot be independently explained.
In simple words, landscape configuration is responsible for the
formation of river networks and river networks are, in turn,
responsible for shaping the landscape, interactively.
This intense and continuous interaction rules the physics of the
overall system (\ie{ the combination of soil and river networks}),
causing:
\begin{itemize}
  \item changes in mass distribution;
  \item energy transfer and transformation. 
\end{itemize}

\blockcquote[see][p. 684]{paik:2011}{
As raindrops fall on a landscape, travelling toward the ocean,
they mobilize surface-forming materials, such as sand and gravel,
and dissipate their energy.
Through this process, fluxes of the sediment-water mixture form
and then move through their respective paths, \ie streams or
rivers. These fluxes exhibit spatially varying mass and energy
distributions.}
These two phenomena will be detailed in the following sections.

%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Changes in mass distribution: the process of matter
movement}
\label{subs:massTransfer}
The first point of view our study of the
phenomena starts from regards mass transfer and transportation: 
firstly their phenomenology is described, then the mathematical
formalization of their main features is given.
%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Phenomenology}
Phenomena \graffito{The water flowing onto the landscape moves
a large amount of soils downstream.} of matter movement which
cause changes in soil mass distribution are usually grouped together under
the term \enquote{solid matter transport}.
In particular, under this category, the following three
sub-processes are grouped:
\begin{itemize}
  \item erosion;
  \item matter transfer;
  \item sedimentation.
\end{itemize}
These three processes take often place sequentially and their
driver is the action of a flowing fluid: a portion of soil is
eroded (\ie removed) by the flow, it is carried by the flow itself
and then settles in a different place from the original one,
through sedimentation.

The prolonged action of this erosion-sedimentation cycle, along
thousands of years, shapes the landscape. In fact, when the water
flows, nourished by raining events, and snow melting start assuming
preferential paths on the hillslopes, the process of river
channels formation and landscape evolution develops according to
the following phases, well described by \citeauthor{liu:2012}
\cite{liu:2012}:
\begin{description}
  \item [Rapid incision:] the flow starts carving the terrain,
  forming V-shaped valleys;
  \item [Widened incision:] flow incision goes on, together with
  landslides and avalanches, widening the valley;
  \item [Dynamic equilibrium:] channels assume a milder slope, the
  rate of sediment from upstream increases and the V-shaped valley
  becomes U-shaped.
\end{description}
The transition from V-shaped to U-shaped valley is well
represented in \myFigNoSpace{fig:VUvalleys}.

\begin{figure}
\myfloatalign
\includegraphics[width=1.0\columnwidth]{Images/VUvalleys.pdf}  
\caption[V shaped and U shaped valleys]{On the left: V-shaped
valley, Riobamba (Ecuador). On the right: U-shaped valley, Glen
Geusachan (UK).}
\label{fig:VUvalleys}
\end{figure}

Moreover, the result of erosion, transport and deposition is the
typical concave shape of a natural river longitudinal profile: as
represented in \myFigNoSpace{fig:riverProfile}, its slope tends to
decrease going from upstream to downstream.

\begin{figure}
\myfloatalign
\includegraphics[width=0.75\columnwidth]{Images/riverProfile.pdf}  
\caption[Example of concave river profile]{An example of concave
river profile: the longitudinal profile of Suceava river
(Romania). $H$ is the stream altitude at the point of measurement,
$H_0$ is the stream altitude from the river mouth at the
headwater, $L$ is the stream distance from the river mouth at the
point of measurement, $L_0$ is the stream distance from the river
mouth at the headwater.\\
Data source: \cite{radoane:2003}.} 
\label{fig:riverProfile}
\end{figure}
		
Considering solid matter transfer in open water channels, it is
possible to say that the dragging action of the flowing current
acts on river bed and banks.
As a consequence, the portion of matter removed can be transported
by the flow alternatively:
\begin{itemize}
  \item as suspended in the flow;
  \item as dragged on the bottom of the channel.
\end{itemize}    
Moreover, going more in detail, a channel has a specific sediment
transport capacity: if the rate of sediment is below this
threshold, no sedimentation happens, but \blockcquote[see][p.
5]{einstein:1950}{if the rate of sediment supply is larger than
the capacity of the channel to move it, the surplus sediment drops
out and begins to cover the channel bottom.}

%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Mathematical modeling} 
Even \graffito{Existing mathematical models focus on different
aspects and processes of the whole system.} if the previous
description of erosion, solid matter transport and sedimentation
appears simplified, it should be already clear that those
processes hide a great complexity, given the number of elements
that take part in them and their interactions.
In fact, they depend again on both: 
\begin{itemize}
  \item factors related to \myEmph{water flows}. They include
  precipitation abundance and its transformation into runoff;
  \item factors related to the \myEmph{soil}. They include shape
  factors, like slope and side length, structural elements (\ie
  kind of soil, texture, grain size and permeability) and other
  factors (like vegetation, soil usage and agricultural cultures).
\end{itemize}

For this reason, great effort was dedicated by hydrogeologists in
providing a mathematical description of erosive processes. In
general the goal is quantifying the rate of erosion and the rate
of solid matter transported.
A complete listing and analysis of the existent formulas and
methods of evaluating them is not the objective of this thesis.
In this section it is just important to understand which factors
are involved and how the process of solid matter transport can be
formalized. Therefore, not all the proposed mathematical
formulations are described here, but only their relevant goals and
some examples, without making the dissertation too
heavy.\footnote{For the reader willing to deepen her/his knowledge
about sediment transport, the following books are suggested:
\fullcite{allen:1985} \cite{allen:1985} and \fullcite{graf:1998}
\cite{graf:1998}.}

Specifically, it is possible to distinguish among different
models, for solid matter movement description, depending on their
target:
\begin{description}
\item[Models for hillslope erosion:] they consider the
mountainside scale and try to link the loss of soil on the
mountainside to the amount of matter which reach the outlet of the
basin. They allow the evaluation of the \textit{Sediment Delivery
Ratio} along a time horizon, as formalized with
 \myEq{eq:SedimentDeliveryRatio}
\begin{equation}
SDR= \frac{P}{A}
\label{eq:SedimentDeliveryRatio}
\end{equation}
where:
\begin{itemize}
  \item $SDR$ is the sediment delivery ratio;
  \item $P$ is the sediment production;
  \item $A$ is the soil loss.
\end{itemize}
Such kind of models can be either physically based, requiring the
use of differential equations, or empirical ones, like the
\textit{Universal Soil Loss Equation} and its newer versions,
which are the most used.\footnote{For a reference, see \fullcite
{wischmeier:1978} \cite{wischmeier:1978}.}
\item[Models for bank erosion:] they are very important in order
to quantify bank erosion and be able to build effective embankments.
Specifically, they usually follow these three steps:
\begin{enumerate}
  \item a threshold parameter $\phi_c$, called critical Shields'
  parameter is evaluated thanks to Shields' abacus, as
  \begin{equation}
	\phi_c= f(Re)
  \label{eq:ShieldsCritical}
  \end{equation}
  where $Re$ is Reynold's number;
  \item $\phi$, which is Shields' parameter for the given river
  and flow, is evaluated as
  \begin{equation}
	\phi= \frac{\tau}{(\gamma_s-\gamma)d}
  \label{eq:ShieldsCriticalBis}
  \end{equation}
  where:
  \begin{itemize}
    \item $\tau$ is the shear stress;
    \item $\gamma_s$ is the specific weight of the sediment;
    \item $\gamma$ is the specific weight of the fluid;
    \item $d$ is the particles diameter.
  \end{itemize}
  \item $\phi_c$ and $\phi$ are compared: if the value of $\phi$
  is over $\phi_c$, bank erosion happens.
\end{enumerate}
\item[Models for matter transfer in riverbed:] their aim is to
quantify the amount of solid matter transported by the water
flowing in the channel.
As already said, it depends both on the characteristics of the
water flow and the solid matter features.
It can be quantified adopting three types of approaches, \ie a
physical based one, a probabilistic one or an empiric one; here,
only the formulation given by the first approach is quoted, in
order to understand the contributing elements.\footnote{For
references on the probabilistic and empiric approach, see,
respectively, \fullcite{einstein:1950} \cite{einstein:1950} and
\fullcite{meyer:1948} \cite{meyer:1948}}

The just mentioned formulation is given by
\citeauthor{duboys:1879} \cite{duboys:1879}, as follows:
\begin{equation}
q_s= n\varepsilon\frac{(n-1)v_s}{2}
\label{eq:DuBoys}
\end{equation}
where:
\begin{itemize}
  \item $q_s$ is the bed material load per unit width;
  \item $\varepsilon$ is the thickness of material of the
  riverbed;
  \item $n$ is the number of layers considered for dividing the
  bottom of the riverbed;
  \item $v_s$ is the velocity of each layer.
\end{itemize}

Moreover, since, as previously said in this section, sediments are
transported both as suspended in the flow and over the river bed,
a more detailed analysis would also evaluate the amount of
suspended sediments as a fraction of the total rate of sediment
load.
\end{description} 

As it was said at the beginning of \mySecNoSpace{sec:basicTheory},
landscape evolution under river dynamics entails two processes
(\ie matter transport and energy transformation). The principal
concepts related to matter transfer were just treated, therefore
it is time to devote the same effort for understanding the ones
related to energy.

\subsection{Energy and entropy-related issues in landscape
evolutionary process}
\blockcquote[see][p. 311]{yang:1971}{The only useful energy nature
provides to a unit mass of raindrops falling on the slope of a
watershed is its potential energy above a datum, say, the sea
level.}
This \graffito{Potential energy of raindrops is converted to
kinetic energy and friction loss. This is the driving force of
erosion.} sentence written by \citeauthor{yang:1971} in his
\citeyear{yang:1971} article, expresses well and simply the
starting point for energy-related analysis when speaking about
landscape evolution connected to river development.
In fact, it is possible to affirm that the potential energy and
its dissipation by friction is the source of landscape evolution.
When raindrops start flowing on the hillslopes following their
way toward the basin outlet, their potential energy is converted
into kinetic energy and friction loss, and the work produced
shapes the terrain with the dynamics described in the previous
sections, forming channels and river networks.
As a consequence, since the current landscape is the result of the
action of raindrops (and consequently rivers) flowing on it for
millions of years, it is also the result of their energy
expenditure.
Starting from this statements, the following concepts become
relevant:
\begin{itemize}
  \item Energy expenditure; 
  \item Entropy;
\end{itemize}
They are going to be treated in the following two paragraphs.

%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy expenditure}
The \graffito{The amount of energy spent by water flows is
assessed by evaluating friction loss and geomorphological work.}
process of potential energy transformation explained in the
previous paragraph is the result of the movement of water flowing
onto the soil, therefore it is driven by the forces allowing the
flow itself.
As \citeauthor{rodriguez:1992} \cite{rodriguez:1992} explain in
their \citeyear{rodriguez:1992} paper, the forces driving water
flows are the gravity force, in one direction, and the the
resistant force due to the soil friction, in the opposite
direction:
\begin{description}
\item[weight:] its component along the slope is the force which
causes the flow of water.
It can be expressed as $F_w$ in \myEqNoSPace{eq:weightForce}, 
\begin{equation}
F_w=\gamma dLwS
\label{eq:weightForce}
\end{equation}
where:
\begin{itemize}
  \item $\gamma$ is the specific weight of the
  fluid;\footnote{$\gamma=9810 N/m^3$ for water with temperature
  of 4°C}
  \item $d$ is the channel depth;
  \item $L$ is the channel length;
  \item $w$ is the channel width;
  \item $S$ is the slope.   
\end{itemize}
\item[resistant force:] assuming as a simplification a rectangular
section of the channel, it can be expressed as
 \begin{equation}
F_r=\tau(2d+w)L
\label{eq:resistantForce}
\end{equation}
where $\tau$ is the stress per unit area and $(2d+w)L$ represents
the wetted area of channel.
\end{description}

Considering the two just mentioned forces, it is possible to
quantify the rate of energy expenditure for a channel having
length $L$ and discharge $Q$.
In fact, when the water flows in the downstream direction, its
initial potential energy moves into kinetic energy while part of
it gets lost because of the interaction with soil and 
resistant friction force.
Assuming that water flows with constant velocity, an
equilibrium between the two forces will exist, therefore $F_w =
F_r$.
Equalling \myEq{eq:weightForce} and
\myEqNoSPace{eq:resistantForce}, one obtains:
\begin{equation}
\tau=\gamma SR
\label{eq:tau}
\end{equation}
being $R = \frac{A}{P_w}$ the ratio between wetted area $A$ and
wetted perimeter $P_w$ \ie the hydraulic radius.
Since, as \citeauthor{rodriguez:1992} state, in the case of
turbulent incompressible flow the boundary shear stress $\tau$ can
be defined as
\begin{equation}
\tau=C_f \rho v^2 
\label{eq:tau2}
\end{equation}
where $C_f$ is a dimensionless coefficient of resistance,
equalling \myEq{eq:tau} and \myEq{eq:tau2}, it is possible to
obtain the rate of energy expenditure per unit weight of fluid per
unit length of channel as expressed in equation
\myEq{eq:unitFrictionLoss}:
\begin{equation}
S=\frac{C_f v^2}{Rg} 
\label{eq:unitFrictionLoss}
\end{equation}
It is now easy to understand that the rate of energy expenditure
for a channel having length $L$ and discharge $Q$ will be:
\begin{equation}
P=C_f \rho \frac{v^2}{R}QL 
\label{eq:energyExpenditurePartial}
\end{equation}
In addition to this expenditure there is a second term
representing the energy expenditure for the \enquote{maintenance
of the channel} \ie the work carried for transporting sediments
and caring the channel \cite{rodriguez:1992}. This second term
depends on a coefficient $K$ related to the soil, on the shear
stress $\tau^m$ with $m$ constant and on the wetted perimeter
$P_w$.
Adding this, completely define the \myEmph{energy expenditure} of
a channel as:
\begin{equation}
P=C_f \rho \frac{v^2}{R}QL + K\tau^m P_wL 
\label{eq:energyExpenditure}
\end{equation}

If some energy concepts related to river dynamics have just been
explained, also the following one, \ie entropy, must be addressed
in order to introduce the concept of minimum energy expenditure
which will be fundamental with respect to overall objective of the
thesis.
 
\subsection{Entropy}
\label{subs:entropy}
The \graffito{Entropy is a concept that can be applied to
river systems in order to improve their understanding.} concept of
entropy in landscape evolution was first introduced by
\citeauthor{leopold:1962} in \citeyear{leopold:1962}, and then
again developed by \citeauthor{yang:1971} in \citeyear{yang:1971}.
Basically, the concept of entropy in river evolution is related to
the same concept in thermodynamics.
There, entropy is defined as follows:
\begin{equation}
\Phi=\int\frac{dE}{T},
\label{eq:entropy}
\end{equation}
being $T$ the absolute temperature and $E$ the thermal energy per
unit mass of substance.

According to \citeauthor{leopold:1962} and \citeauthor{yang:1971},
it is possible to imagine an analogy between thermodynamics and
river elements:
\begin{itemize}
  \item elevation $Z$ in river evolution assumes the same role of
  the temperature in thermodynamics;
  \item potential energy loss $H$ in river evolution assumes the
  same role of the thermal energy in thermodynamics.
\end{itemize} 
It becomes licit to redefine \myEq{eq:entropy} as follows:
\begin{equation}
\Phi=\int\frac{dH}{Z_m} 
\label{eq:entropyForRivers}
\end{equation}
being $Z_m$ the average total fall from the beginning of first
order stream to the end of the $m$-th order stream.\footnote{The
ordering procedure to classify river branches will be introduced
in \mySubsecNoSpace{subs:hortonsLaw}. Here is enough to know that
the streams originating from springs have order 1 while streams
closer to the outlet of a river have higher order.}

Basing their analysis on this analogy, the mentioned authors
assert that a system moves toward stationarity with a minimum rate
of production of entropy. As a consequence, it is possible to
write \myEq{eq:minEntropy}, which represents the concept behind
the one of minimum energy expenditure in river evolution deepened
in \mySec{sec:lap}.\footnote{The term \enquote{minimum} has to
be intended as the minimum compatible with the external conditions.}
\begin{equation}
\frac{d\Phi}{dt}=\text{minimum} 
\label{eq:minEntropy} 
\end{equation} 

\section{State of the art analysis approaches}
\label{sec:approaches}
Now \graffito{Studies about landscape evolution try both to
understand the system processes and to provide reliable management
tools.} that the main drivers and features of the complex
phenomenon of landscape evolution under river dynamics should be
clear, some questions rise:
\begin{enumerate}
  \item Who is interested in studying this evolutionary process?
  and why?
  \item How is it possible to describe it in a complete way?
\end{enumerate}

\subsection{Goals intended to be reached}
The answer to the first question should be accurately balanced,
since it constitutes part of the motivation for having carried out
this work and, also, the perspective for future research or work
this thesis would be addressed to.
Fortunately, the cross analysis of the existent literature helped
the authors of this thesis to provide an answer to this question.
It is possible to say that two kinds of interests drive the
research on the considered topic: scientific and technical.

In fact, the pioneers of this field of study were hydrologists and
geomorphologists who were attracted by the regularity of some
patterns in natural landscapes and started to study landscape and
river evolution \enquote{for the sake of knowledge}.
The first interest was then focused on understanding the processes
going on in the nature, in order to find some general laws
formalizing them and produce descriptive models.
Any further improvement in these kind of models would enrich the
knowledge available to the scientific community.

On the other side, the development of technical disciplines
related to goals such as management of natural resources
(especially water) or soil protection needs models reproducing
landscape and river evolution. With this kind of tools, the
following issues can be exploited:
\begin{itemize}
  \item to forecast the dynamics of a landscape and support
  decision making processes in planning strategies and
  localization of infrastructures (such as dams, artificial river
  banks, and so on);
  \item to help the forecast of the landscape response and resilience following
  extreme events;
  \item to better know and manage the availability of resources of
  a territory.
\end{itemize}  
If the interests in studying landscape and river evolution appear
to be many, the second question still remain unsolved. So, how is
it possible to describe that processes in a complete way?

\subsection{Methodologies to reach the goals}
The answer is neither unique nor simple. In scientific and
technical literature on the topic, different approaches
to study landscape and river evolution processes can be found. They differ by
dimension of the study domain, by scale of the processes faced, by
goals achieved ans so on.

In particular, the following different categories of approach were
identified to better explain the main differences:
\begin{itemize}
  \item physically-based approach;
  \item optimality-based approach and probability-based approach.
\end{itemize}

\subsection{Physically-based approach}
\label{subs:physicalModels}
Physically \graffito{At the scale of river sections, this
approach allows to reproduce in a model the state of the art
knowledge of the processes.} based models approach the topic of
landscape evolution under river dynamics with a descriptive
perspective.
They try to describe the current physical processes \ie water flow
and matter transfer, and then to reproduce them using the
parameters through which they are characterized.
They consider in the most complete way as possible the processes
and the elements involved in water and matter movement.

Usually, as also \citeauthor{pazzaglia:2003} asserts,
\blockcquote[see][p. 257]{pazzaglia:2003}{Physical models
typically isolate one part of the geomorphic system}, therefore
they are usually built on a smaller scale than the one of a river
network. They focus on parts of a river channel, between sections as the
one represented in \myFigNoSpace{fig:riverSection}, to study what
happens in terms of water and matter movement when the flows goes
from the upstream one to the downstream one.
\begin{figure}
\myfloatalign
\includegraphics[width=1.0\columnwidth]{Images/riverSection.pdf}  
\caption[Sketch of river section]{Sketch of river section.}
\label{fig:riverSection}
\end{figure}

To do this, they make use of physical laws, when they are known, and
fill the unknown with empirical relations that contains parameters
as the ones quoted in the equations mentioned in
\mySubsecNoSpace{subs:massTransfer}.
The best example for that is given by \citeauthor{nanson:2008}
in the following three equations \cite{nanson:2008}:
\begin{description}
\item[flow continuity:]
\begin{equation}
Q=WDV
\label{eq:flowContinuity} 
\end{equation} 
\item[flow resistance:]
\begin{equation}
V=\sqrt{\frac{8}{f}gRS_f}
\label{eq:flowResistance} 
\end{equation}
\item[sediment flux:]
\begin{equation}
\frac{Q_s}{W}=c_d(\tau_0 - \tau_c)^{1.5-1.8}
\label{eq:sedimentFlux} 
\end{equation} 
\end{description}
Looking at the equations it is possible to identify parameters
related both to the water flow, channel section and soil
characteristics, but especially to their interaction.
In fact, there is:
\begin{itemize}
  \item channel width $W$, average depth $D$, hydraulic radius $R$
  and slope $S_f$ describing the section;\footnote{with
  $D=\frac{A}{W}$ being $A$ the wetted area (blue in
  \myFigNoSpace{fig:riverSection}), $w$ the width of the channel
  measured at the top of the section, and $R=\frac{A}{P}$ being
  $P$ the wetted perimeter.}
  \item average flow velocity $V$ and discharge $Q$ characterizing
  the water flow;
  \item friction factor $f$ and a coefficient $c_d$ linked to the
  grain size of the soil;
  \item flow carrying capacity for sediment discharge $Q_sc$,
  shear stress $\tau_0$ and critical shear stress $\tau_c$
  characterizing the interaction between the fluid component and
  the solid one.
\end{itemize}

\subsubsection{Strengths and limits of physically-based approach}
The \graffito{Physically-based approach allows deep
understanding of sub-processes in landscape evolution but fails to
deal with basin and network scale modeling.} physical base of this
approach allows to build descriptive and management models that
are justifiable and reliable. The process of landscape evolution
is rationally divided into sub processes, which are analyzed and
explained and eventually recomposed into the whole system by
explicitly studying their interrelations.
The physical base also means that the quantities involved have a
real counterpart, which can be measured directly to calibrate the
models. The goal achieved is a scientific and rational
understanding of the phenomena involved in landscape evolutions.

On the other side, the use of this approach at the basins scale 
is not common, even if possible. As \citeauthor{pazzaglia:2003}
says, \blockcquote[see][p. 257]{pazzaglia:2003}{Physical models
[\ldots] rarely show the behavior of an alluvial channel, for
example, in the context of broader landscape change}. Models based
on physical approach are usually applied on a segment of a river
channel between two sections, whose geometry and 
characterizing parameters must be measured. Recalling
\myEq{eq:energyExpenditure} makes clear the difficulties related
to the number of parameters.

\begin{equation}
P=C_f \rho \frac{v^2}{R}QL + K\tau^m P_wL 
\label{eq:energyExpenditureBis}
\end{equation}

A model applying this equation needs a value for $C_f$ and $K$,
the geometry of the river for each section the simulation is performed 
into and a coupled hydraulic model to evaluate
$v$, $P$, $R$ and $Q$. In the absence of measures for these
quantities, a mathematical calibration can be challenging
because of the range they can assume. Also, if any of these values
is changing over time, the model needs to know it. Moreover its
hydraulic part and the geometry of the channel are
interconnected and, therefore, the additional degrees of freedom
must be tackled with extra relations.

This characteristics heavily limited the use of physical based
approach in landscape evolution models at basin
scale.\footnote{Although one example can be found in
\cite{peckham_mathematical:2003} and its bibliography references.}

Before introducing the second category of study approach, \ie 
optimality-based approach and probability-based approach, some 
other concepts are introduced in the next paragraphs.

\section{The idea of a least action principle}
\label{sec:lap}
It \graffito{The optimality based approach is the idea guiding
researchers that deal with landscape evolution at a basin
scale.} has been shown the strength of the physical approach to
understand landscape evolution, but also its limitation, comes when 
modeling at basin scale. It should be underlined that the
scale and choice of modeling approaches closely reflect the
targets of the research performed, \eg to study river system
resilience to climatic changes. For such a goal, \myEq{eq:energyExpenditure} 
needs to be deeply simplified.

This process of reduction lead \citeauthor{leopold:1962}
\cite{leopold:1962}, \citeauthor{yang:1971} \cite{yang:1971},
\citeauthor{rodriguez:1992} \cite{rodriguez:1992} and many others
to deepen the study of energy expenditure and minimum entropy at
river basin scale. They came up with different mathematical
formalizations of an optimality principle, in accordance with the
specific topic of their research. But they share one clear idea:
river systems follow a kind of optimality.

This idea will be clarified in the following sections.

\subsection{Least action principle}
Given \graffito{Reductions of physical laws to scale up the study
of river systems at a basin level lead to the idea of an optimal
principle.} the fact that, according to \citeauthor{leopold:1962}
and \citeauthor{yang:1971}, \myEq{eq:minEntropy} represents the
law governing the evolution of a system toward its most probable
state, it is also possible to state the following sentence:
\blockcquote[see][p. A7]{leopold:1962}{The river channel has the
possibility of internal adjustment among hydraulic variables to
meet the requirement for maximum probability, and these
adjustments tend also to achieve minimization of work.}
This minimum work condition is also called \myEmph{\acl{LAP}}
\acs{LAP}.

Theories which support the \ac{LAP} affirm the idea that a river,
and consequently the landscape, evolves in a way that its energy
expenditure is minimum.
Nevertheless, there is disagreement on how to traduce in
mathematical terms the concept of minimum energy expenditure when
speaking of river evolution, since \myEq{eq:energyExpenditure} may
be reduced in alternative ways according to the research target,
\eg if intended to be applied to the network as a whole or to
study the three dimensional structure of a landscape.
As a consequence, scientists have been focusing on the research
for a proper formalization of the criterion to be minimized.
The result of their activity is a great number of proposed
criteria expressing the \ac{LAP} with different perspectives. So
far, none of them is satisfactory to fully reproduce the
complexity of a river network while capturing its 3D structure.

Many of them were listed and explained by \citeauthor{paik:2010}
in \citetitle{paik:2010} and by \citeauthor{paik:2012} in
\citetitle{paik:2012}. The ones closely related to this work will
be shown below.

\subsubsection{Stream Power}
A first example of an optimal principle as a reduction of the
physical laws is the so called \textit{Unit Stream Power},
formulated by \citeauthor{yang:1985} \cite{yang:1985} directly
from \myEq{eq:minEntropy} as:
\begin{equation}
\frac{dY}{dt}=VS=\text{minimum} 
\label{eq:unitStreamPower} 
\end{equation} 
where:
\begin{itemize}
  \item $V$ is the average flow velocity;
  \item $S$ is the slope.
\end{itemize}
and entropy $\Phi$ is substituted with $Y$, the average fall of a
stream, since they are proportional.

\subsubsection{Principles of energy expenditure}
We are now ready to present the optimal principle used in this
thesis. It has been proposed by \citeauthor{rodriguez:1992} in
\citetitle{rodriguez:1992}, a paper that has become a milestone of
the discipline and is the reference for other studies.\footnote{It
has almost $180$ citations according to Google Scholar and $142$
according to Web Of Knowledge}

In the paper, \citeauthor{rodriguez:1992} enunciate three
specifications of the \ac{LAP} that, according to their theories,
are respected by river networks.
\begin{description}
	\item[minimum energy expenditure in any link] According to this
	principle, energy expenditure is at its minimum in any link of
	the river network.
	\item[equal energy expenditure] According to this principle, the
	energy expenditure relative to a unit area is constant anywhere
	in the network.
	\item[minimum total energy expenditure] This principle expresses
	the idea that the sum of the energy expenditure of all the links
	in a network should also be a minimum.
\end{description}

The paper authors used this three principles to identify a
mathematical formalization of the \ac{LAP} they used to describe
the planar structure of river networks. The second principle
ensures that
\begin{equation}
P_1 = \frac{P}{P_w L} = \text{const}
\label{eq:secondPrinciple}
\end{equation}
and being $v = \frac{QP_w}{A}$ and $\tau = C_f \rho v^2$,
\myEq{eq:energyExpenditure} becomes
\begin{equation}
P_1 = C_f \rho v^3 + K C_f^m \rho^m v^{2m} = \text{const}
\label{eq:energyExpenditureAndSecondPrinciple}
\end{equation}
which means the velocity is constant throughout the network, 
being constant all the other quantities in the equation.

Substituting the width $w = \frac{Q}{vd}$ and grouping all the
constants in \myEqNoSPace{eq:energyExpenditureAndSecondPrinciple},
the following expression is obtained:
\begin{equation}
P=\frac{QL}{d}\times \text{const} + dL\times \text{const}
\label{eq:energyExpenditure1stPrinciple} 
\end{equation} 
According to the first principle, in a link of the network $P$
must be at its minimum. Therefore, it should be that
$\frac{dP}{d(d)}=0$, which implies $Q\propto d^2$.
Substituting this result in
\myEq{eq:energyExpenditure1stPrinciple}, the optimal energy
expenditure can be rewritten as:
\begin{equation}
P=kQ^{0.5}L
\label{eq:optimalEAnyLink} 
\end{equation} 
Finally according to the third principle, the energy expenditure
in the river network as a whole should be a minimum, \ie the sum
over the $N$ links of the network of their energy expenditure
should be
\begin{equation}
P=k\sum_{i=1}^N{Q_i^{0.5}L_i} = \text{minimum}
\label{eq:optimalETotal} 
\end{equation} 

It \graffito{The different formulations of the \ac{LAP} are in
contrast one another.} is possible to comment that, even
if all the three principles express the \ac{LAP}, they are not
equal, \ie the optimization of one of them not necessarily implies
the optimization of the others. This is stated in the paper by
\citeauthor{rodriguez:1992} \cite{rodriguez:1992} and suggests the
idea that these principles captures opposite \enquote{parts} of
the \ac{LAP}. As we will see in the next chapters, this opposition
is called conflict in the system management field which usually
deals with opposite criteria. The meaning is that the fulfillment
of one of them is in contrast with the fulfillment of one or
more of the others.
This point confirms the presence of different proposed criteria
and the debate around the different listed principles of minimum
energy expenditure is so open.

In the next paragraph the applications of these principles are
shown and contextualized. The reader will be guided through the
application of optimal principle in landscape evolution. We will
assess different methodological approaches retrieved from the
literature, in order to introduce and clarify the purpose of this
thesis.

\subsection{Probability-based approach}
According to this category of approach, landscape and rivers
evolve in order to move toward their most probable state. The
definition of most probable state is close to the concept of
optimality, therefore we have listed this approach here.

The first, and probably most important work in this direction is
represented by the one carried out by \citeauthor{leopold:1962} in
\citeyear{leopold:1962}.
In their paper, the authors assert that river networks
self-organize in order to reach the configuration with the most
probable distribution of energy, which, as shown in
\mySubsec{subs:entropy} \blockcquote{leopold:1962}{represents
maximum entropy}.

In particular, they simulated a random-walk model for reproducing
the longitudinal profile of a river. It showed that, on average,
the profiles obtained as the output of the model have a concavity
comparable to the one visible in natural rivers. They also
simulated another constrained random-walk model for reproducing
the 2D pattern of an entire river network. They assert that
\blockcquote[see][p. A15]{leopold:1962}{random pattern represents
a most probable network in a structurally and lithologically
homogeneous region}.

Despite the results, as \citeauthor{perron:2012} states,
\blockcquote[see][p. 100]{perron:2012}{Such studies have provided
new insights into the structure of natural river networks, but
cannot directly relate river network form to the erosional
mechanisms that shape the topography}.

\subsection{Optimality-based approach}
\label{subs:optimalityApproach}
Approaches based on the idea of the \ac{LAP} more than the
probability based one, assume that river networks evolve
according to a particular criterion corresponding to the
minimization of a certain mathematical formulation.
Accordingly, river and landscape weathering can be simulated by
solving a minimization problem, also called optimization problem.
The choice of the criterion to be optimized becomes the initial
assumption. Then, the comparison between natural river networks and
optimized ones verifies the correctness of this initial
assumption. The smaller scale phenomena \eg rate of erosion,
channel dimensions, grain size dimension and others, are not
detailed as in the physical-based approach. The only thing
that matters is the configuration of the landscape and the network
above it to quantify the selected criteria.

\subsubsection{Total energy expenditure and \acs{OCN}}
One important work within this approach is represented by the
already mentioned \citetitle{rodriguez:1992} by
\citeauthor{rodriguez:1992} and the subsequent \cite{rigon:1993}.
They tried to test the principle of minimum total energy
expenditure showed in \myEq{eq:optimalETotal} on a $2D$ network,
in the following way:
\begin{aenumerate}
  \item an initial network is perturbated with random noise;
  \item step 1 is repeated many times, in order to have many
  sample networks;
  \item for each sample network the value \acs{TEE} of total
  energy expenditure is computed as in \myEq{eq:optimalETotal}
  \begin{equation}
  P=k\sum_{i=1}^N{Q_i^{0.5}L_i}
  \label{eq:optimalETotalBis} 
  \end{equation}
  \item the network with minimum value of $E$ is selected and
  evaluated according to Horton's bifurcation ratio and length
  ratio indexes.\footnote{These indexes are common measure of
  river networks. For a wide description of Horton ratios and
  other hydrological indexes, see
  \myChap{chap:testingOptimality}.}
\end{aenumerate}

The results of the combination of the three principles of energy
expenditure into \myEq{eq:optimalETotalBis} results in a unified
picture of the most empirical findings related the structure of
river networks. In fact, \blockcquote[see][p.
1645--1646]{rigon:1993}{it is concluded that a comprehensive
framework for the investigation of geophysical structures,
different from their mere geometrical description or from modeling
of the growth process responsible for their construction is
achieved through \ac{OCN} approaches.} \blockcquote[see][p.
2194]{rinaldo_minimum:1992}{Also, the geomorphological description
of \acp{OCN} has been studied in detail, revealing an almost
perfect match with well-known empirical or experimental result.}

Nonetheless, as recognized by \citeauthor{paik:2011}
\cite{paik:2011}, the approximation of a landscape with a 2D
network lacks the direction gravity acts on. The
good results from \acp{OCN} were possible only because the
elevation, even if is not explicitely included into \ac{TEE}
(\myEq{eq:optimalETotalBis}) it is considered by the river network setup, which
organizes values of $Q$ according to it. 
In order to develop a framework to fully study
river networks and landscape features, it is required to
completely include the third dimension: elevation.

\subsubsection{\acl{GLE}}
After the previously described work, another important step in exploring this modeling
approach is constituted by the work performed by
\citeauthor{paik:2011} in \citeyear{paik:2011} \cite{paik:2011},
who tries to include the third dimension \ie elevation, in order
to extend the application of the approach.
As in the previous case, a single objective is considered:
the one of minimum \ac{TEE}, defined as the
above \myEqNoSPace{eq:optimalETotal}.
 
\citeauthor{paik:2011} comments his results saying that
\blockcquote[see][p. 690]{paik:2011}{simulation results
[\ldots] exhibit a self-similar tree structure of natural river
networks}, but \blockcquote[see][p. 690]{paik:2011}{Optimized
river networks show no pattern in their longitudinal profile}, so
the description of the third dimension remains still partial.
In fact the longitudinal profile of a river shows a typical
concave feature as in \myFig{fig:riverProfile}. The profile should
be steeper for highest elevations and more flat as the river
approaches its delta. Missing this characteristic from
\citeauthor{paik:2011}'s results means that both (or either) the
model and the proposed criteria are not able to fully understand
the complexity of the evolutionary process.

\section{A multi-objective framework}
\label{sec:MOframework}
Given the theoretical framework and the premises explained in the
previous sections, it is necessary now to clarify the objective of
this thesis, already mentioned in the introduction.
The general, \myEmph{overall objectives} of disciplines studying
landscape and river evolution are to try to accurately describe
that process and being able to reproduce it with modeling tools.
As a sub-objective, we identify a more \myEmph{specific
objective}:
testing optimality principles followed by whole landscape
evolution through a multi-objective approach based on a 3D
representation.

In fact, it appears from the assessment of the existent study
approaches, performed in \mySec{sec:approaches}, that optimality
principles approach is promising in terms of scalability and can
be applied successfully to the whole basin.
It seems so that as for the methodology, the approach based on
optimality criteria represents a good compromise between accuracy
and complexity.
Nevertheless, as it is also confirmed by the considered
literature, at present, a single, satisfactory, formulation of the optimal principle
has not been formulated yet: the already proposed optimality
criteria, as they are considered separately, partially describe
the process \eg \citeauthor{paik:2011}'s proposed \ac{TEE}
criterion was not able to make his model reproduce the typical
concave profile of main network channels.
Given the previous considerations, the question we would address
is: may a multiple-objective approach explain more exhaustively
the considered process of landscape evolution?

This does not imply that landscape and river networks evolve
according to multiple criteria; instead, it means that, given the
proven limits of single-objective models, a multi-objective
approach would be a more complete analysis tool and help to
find the formulation of the optimal principle followed by nature.

In order to pursue the declared specific objective, a framework
for testing multiple optimality criteria was built by the authors
of this thesis.
It will be described in detail in the following chapters, but
before entering the technical issues, the optimality criteria
which will be considered are now listed and defined.

\subsection{Optimality principles: \acs*{TEE}, \acs*{EEL},
\acs*{EE}, \acs*{EEE}}
\label{subs:optimalityPrinciples}
Four optimality principles have been chosen to be minimized as
objectives in the multi-objective framework analysis that will be
described in the next chapters.
The four objectives were chosen mainly on the basis of the
already cited \citeauthor{rodriguez:1992}'s and
\citeauthor{paik:2012}'s articles \cite{rodriguez:1992}
\cite{paik:2012}.
Each of them is now described and the reason for the choice is
justified:

\begin{description}
\item[Minimum total energy expenditure]
It represents the third criteria expressed in
\cite{rodriguez:1992}, as already explained in \mySubsec{sec:lap}.
It is the same objective used by \citeauthor{paik:2011} in
\cite{paik:2011}.
It express the minimization of energy expenditure by the river
network as a whole.
Its formulation is obtained from the criterion expressed
in \myEq{eq:optimalETotal}:\footnote{Constant $k$ does not affect
minimization, therefore is not reported in \myEq{eq:TEE}}
\begin{equation}
\highlight{\text{TEE}=\min\left(\sum_{i=1}^N{Q_i^{0.5}L_i}\right)}
\label{eq:TEE} 
\end{equation}  
where $N$ is the total number of links of the network.
It has the following units:
\begin{equation}
\text{TEE}[=]\sqrt{\frac{m^3}{s}}m
\label{eq:TEEunits} 
\end{equation} 
This criteria has proven to be able to reproduce the 2D structure
of river networks in \cite{rodriguez:1992}, \cite{rigon:1993} and
\cite{paik:2011}.

\item[Minimum energy expenditure in any link]
Since \citeauthor{rodriguez:1992} \cite{rodriguez:1992} state
that the energy expenditure should be minimum not only in the
network as a whole, but also in any link of the network, this
second objective is introduced:
\begin{equation}
\highlight{\text{EEL}=\min\left(\var(\mathbf{Q^{0.5}L})\right)}
\label{eq:EEL} 
\end{equation}  
where $N$ is the total number of links of the network and
$\mathbf{Q^{0.5}L}$ is the vector containing the product
${Q_i^{0.5}L_i}$ for every $i$-th link, with $i\in[1:N]$. Being
the minimization of a variance, it is dimensionless.

This objective has to be considered jointly with \ac{TEE}: by
itself it does not minimize the energy expenditure in any link of
the network, but, by imposing that the \ac{TEE} should be
distributed equally in the network, it helps reaching the minimum
in any link. On the other hand, it also means that each link of
the network should be able to reach the same level of energy
expenditure.

\item[Minimum energy expenditure per unit area]
This third objective corresponds to the minimization of the
energy expenditure per unit area in the whole network.

Rewriting the energy expenditure per unit area of the channel
$P_1$ in \myEq{eq:energyExpenditureAndSecondPrinciple} as
\begin{equation}
P_1 = \frac{\rho g Q S}{w + 2d} + K C_f^m \rho^m v^{2m}
\label{eq:energyExpenditurePerUnitArea}
\end{equation}
and recalling that the first principle $\frac{dP}{d(d)}=0$ implies
$d \propto Q^{0.5}$ and $w \propto Q^{0.5}$, one can write
\begin{equation}
P = c_1 Q^{0.5} S + c_2.
\end{equation}
For a given flow condition $c_1$ and $c_2$ are two
constants and therefore
\begin{equation}
Q^{0.5} S = \text{const}
\label{eq:optimalEequal} 
\end{equation}
Principles of this form have been found also by studies on stable
sections in rivers where is stated the idea that this quantity
should be also a minimum \cite{nanson:2008}.

Finally we are ready to express the \ac{EE} as follows:
\begin{equation}
\highlight{\text{EE}=\min\left(\sum_{i=1}^N{Q_i^{0.5} S_i}\right)}
\label{eq:EE} 
\end{equation}  
where $N$ is the total number of links of the network.
It has the following units:
\begin{equation}
\text{EE}[=]\sqrt{\frac{m^3}{s}}
\label{eq:EEunits} 
\end{equation}
It is hoped that the explicit introduction of slope into the
objective formulation helps the generation of concave profiles
for the main channels of the network.

\item[Equal energy expenditure per unit area]
This objective interprets the principle of equal energy
expenditure proposed by \citeauthor{rodriguez:1992}
\cite{rodriguez:1992}.
Since, as written in \myEq{eq:optimalEequal} it should be
verified that $Q_i S_i=\text{const}$ over the network links, the
same condition can be imposed by writing:
\begin{equation}
\highlight{\text{EEE}=\min\left(\var(\mathbf{Q^{0.5}S})\right)}
\label{eq:EEE} 
\end{equation} 
where $\mathbf{Q^{0.5}S}$ is the vector containing the product
${Q_i^{0.5}S_i}$ for every $i$-th link, with $i\in[1:N]$.
Again, since the objective is the minimization of a variance, it
is dimensionless.
As for \ac{TEE} and \ac{EEL}, this objective should be considered
together with \ac{EE}. It is closer to the translation of a
constraint into an objective than an objective by itself.

\end{description}

Wrapping up the content of this chapter, the basic knowledge about
landscape evolution under river dynamics was provided. Then the
principal methods for analyzing this phenomenon are assessed and
it has been presented the framework proposed in this thesis,
together with its aim and the objectives considered. The
formalization of the mentioned framework is going to be explained
in the next two chapters.
